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Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System

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Abstract

This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive, while the one for dephosphorylation is distributive (or vice versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism; if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh–Hurwitz criterion, a Hopf bifurcation criterion due to Yang, and a monomial parametrization of steady states.

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Notes

  1. Network (1) is symmetric to the mixed-mechanism network in which phosphorylation is distributive (instead of processive) and dephosphorylation is processive (instead of distributive), so our results apply equally well to that network (cf. Suwanmajo and Krishnan 2015, networks 21, 22).

  2. This file and others mentioned below are in the Supporting Information; see “Appendix A”.

  3. The functions are provided as a text file in the Supporting Information. See “Appendix A”.

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Acknowledgements

AS was partially supported by the NSF (DMS-1312473/1513364 and DMS-1752672) and the Simons Foundation (#521874). AS thanks Jonathan Tyler for helpful discussions. CC was partially supported by the Deutsche Forschungsgemeinschaft DFG (DFG-284057449).

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Authors and Affiliations

  1. HTW Berlin, Berlin, Germany

    Carsten Conradi

  2. Northern Illinois University, DeKalb, USA

    Maya Mincheva

  3. Texas A&M University, College Station, USA

    Anne Shiu

Authors
  1. Carsten Conradi
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  2. Maya Mincheva
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  3. Anne Shiu
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Correspondence to Maya Mincheva.

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Electronic supplementary material

A Files in the Supporting Information

A Files in the Supporting Information

The following files can be found as supplementary material:

Text files:

  • mixed_H5N_kb.txt ...contains H5N, the numerator of \(\det H_5\) under the assumption \(k_2=k_6=k_9=k_b\)

  • mixed_W.txt ...contains a matrix W that defines (3)

  • mixed_xt.txt ...contains xt, the parameterization (7)

  • mixed_Jx.txt ...contains Jx, the Jacobian evaluated at the parameterization (7)

Mathematica Notebooks:

  • mixed_analysis_H5N_x1_LT.nb:

    Functionality: This file can be used to obtain \(\text {numerator}(\det H_5)\) as in (12), in particular to examine the coefficients \(\alpha _{01}\), \(\alpha _{10}\), ...

    Input: the file mixed_H5N_kb.txt

  • mixed_analysis_H5N_x2_LT.nb:

    Functionality: This file can be used to obtain \(\text {numerator}(\det H_5)\) as in (14), in particular to examine the coefficients \(\alpha _{0}\), ..., \(\alpha _{3}\) and \(\beta _0\), ..., \(\beta _3\).

    Input: the file mixed_H5N_kb.txt

  • mixed_coeffs_charpoly.nb:

    Functionality: This file can be used to obtain the characteristic polynomial of the Jacobian of the system (2). It contains the Mathematica commands to establish \(b_i>0\).

    Input: the file mixed_Jx.txt

  • mixed_Hi.nb:

    Functionality: This file can be used to obtain the determinants of the Hurwitz matrices \(H_2\), ..., \(H_5\). It contains the Mathematica commands to establish \(\det H_i >0\), for \(i=2\), 3, 4 and that \(\det H_5\) is of mixed sign.

    Input: the file mixed_Jx.txt

  • mixed_generate_rc.nb:

    Functionality: This file contains a realization of Procedure 5.1.

    Input: the files mixed_H5N_kb.txt, mixed_W.txt, mixed_xt.txt, mixed_Jx.txt.

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Conradi, C., Mincheva, M. & Shiu, A. Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System. Bull Math Biol 81, 1829–1852 (2019). https://doi.org/10.1007/s11538-019-00580-6

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  • DOI: https://doi.org/10.1007/s11538-019-00580-6

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